Manchester City are contemplating whether to make a third offer for Raheem Sterling and may have to make him the most expensive signing of the Abu Dhabi era after another unsuccessful attempt to persuade Liverpool to sell the England international, this time with a financial package of £40m.

City’s second bid comprised an initial £35m plus another £5m in potential add-ons and was a £10m increase on their initial offer last week, signifying the club’s determination to avoid what is already a sensitive issue at Anfield from dragging on all summer.

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At the same time, the deposed Premier League champions have been informed they will have to increase their offer to Fulham for the England under-19 international Patrick Roberts. The club have made a £5m offer for the 18-year-old but Fulham want considerably more and City will have to go back with a higher offer to make the deal happen.

A compromise is likely to be reached in both cases but City’s information from Liverpool is that the asking price is around the £50m mark and that would mean making Sterling their most expensive acquisition since Sheikh Mansour bought the club in 2008. The previous high was set in 2009 when Carlos Tevez joined in a deal that was initially described as being worth £25.5m, only for it to emerge it had actually cost the club about £47m.

That was in an era when City largely went by a money-no-object rule but they have modified their approach over recent seasons and made it clear on several occasions that they will not pay what they consider to be vastly inflated prices just because selling clubs know of the wealth that exists among the sheikh and his colleagues at the Abu Dhabi United Group. That stance is now being severely tested.

Liverpool’s public position is that they do not want to sell Sterling, who has two years remaining on his contract, but the situation has deteriorated since he turned down a new £100,000-a-week deal in January and diplomatic relations were called off with his agent, Aidy Ward, after an outspoken attack on the club when he said his client would not even sign for “£900,000 a week”.

Ward has made it clear he will lead an aggressive attempt to force a transfer and Liverpool have been so dismayed by his approach that they cancelled a meeting that had been arranged in May with the idea of reopening contract talks.

Liverpool’s sixth-placed finish last season means they cannot offer their player Champions League football next season, whereas City can provide a place in Europe’s leading club competition, an improved chance of silverware and also a considerably higher salary. There is no doubt Sterling would like the move to go ahead and, in City’s favour, they are aware that Liverpool are trying to raise money for their own transfer targets. Christian Benteke is at the top of the list, with a buyout clause in his contract at Aston Villa set at £32.5m. Liverpool have also made an unsuccessful opening bid of £10m for Southampton’s Nathaniel Clyne.

City are deliberately targeting English players to help them comply with the homegrown rules. Sterling is regarded as a better option than Samir Nasri while Roberts has been identified as a potential star of the future. The teenager has turned down a new contract at Craven Cottage and if a deal can be arranged it is likely he will be loaned out next season. Fabian Delph of Aston Villa is another possible target, although not a priority.

As it stands, there are only three players in City’s senior squad who qualify as homegrown – Joe Hart, Gaël Clichy and Richard Wright, who has not played a first-team game since he was with Ipswich in 2011. James Milner has signed for Liverpool while Frank Lampard, Scott Sinclair and Dedryck Boyata have all been released or sold and Micah Richards has just agreed a four-year deal with Villa on a free transfer now his contract at City has expired.